You don't need your account on your old server. When you switch to a new server you change the "DNS" records to tell the world that your site is now at the new server. Once that is done (and it has propogated throughout the internet - which can take a couple of days), you will have absolutely no further need for your old server. Your new host will be able to help you with that process, but it is very easy.
301 redirects are also easy to do. You just basically list the old URL (without the domain.com bit) and the new URL (with the domain.com bit). You put these into the .htaccess file which will be at the root of your wordpress installation.
A redirect looks like:
Code:
redirect 301 /old-page/ http://domain.com/new-page/
It's simple really. It's easiest just to try it and see what happens! Just try it with one of your existing pages (or make up a couple of pages to play with).